3.1.64 \(\int (1-x)^{99} x \, dx\) [64]

Optimal. Leaf size=23 \[ -\frac {1}{100} (1-x)^{100}+\frac {1}{101} (1-x)^{101} \]

[Out]

-1/100*(1-x)^100+1/101*(1-x)^101

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Rubi [A]
time = 0.00, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \begin {gather*} \frac {1}{101} (1-x)^{101}-\frac {1}{100} (1-x)^{100} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - x)^99*x,x]

[Out]

-1/100*(1 - x)^100 + (1 - x)^101/101

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\int \left ((1-x)^{99}-(1-x)^{100}\right ) \, dx\\ &=-\frac {1}{100} (1-x)^{100}+\frac {1}{101} (1-x)^{101}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(567\) vs. \(2(23)=46\).
time = 0.00, size = 567, normalized size = 24.65 \begin {gather*} \frac {x^2}{2}-33 x^3+\frac {4851 x^4}{4}-\frac {156849 x^5}{5}+627396 x^6-10217592 x^7+140066157 x^8-1654114616 x^9+\frac {85600431378 x^{10}}{5}-157366449604 x^{11}[phone] x^{12}-9696194317908 x^{13}[phone] x^{14}-\frac {2062057324941768 x^{15}}{5}+\frac {4750096337812287 x^{16}}{2}-12666923567499432 x^{17}+62806829355518017 x^{18}-290505891817783026 x^{19}+\frac {12572449429225165403 x^{20}}{10}-5104603527655330314 x^{21}+19490304378320352108 x^{22}-70132813684308016488 x^{23}+238292173768273828749 x^{24}-\frac {19146258135816088501224 x^{25}}{25}+2331916055003241548226 x^{26}-6736646381120475583764 x^{27}+18488763007525700846649 x^{28}-48264408157576669898532 x^{29}+\frac {599857644244167183024612 x^{30}}{5}-284248449886557530554488 x^{31}+\frac {2570079734390957672096829 x^{32}}{4}-1386787891870780679371896 x^{33}+\frac {5720500053966970302409071 x^{34}}{2}-\frac {28206275157871771317939099 x^{35}}{5}+\frac {42585944846198556695711973 x^{36}}{4}-19237666204653402059452899 x^{37}+33300287699283081927474024 x^{38}-55246631151825154632274992 x^{39}+\frac {439428796464188236515924114 x^{40}}{5}-134109601422466453670901168 x^{41}+196374773511468735732390996 x^{42}-276016272695076305811040776 x^{43}+372502480574415750374856978 x^{44}-\frac {2414047083412492769871166152 x^{45}}{5}+601126348833940887359223192 x^{46}-719077854633508484642475024 x^{47}+826548729646668720118931889 x^{48}-913043842041304917057126672 x^{49}+\frac {24233705307512968006891237086 x^{50}}{25}-989130828878080326811887228 x^{51}+970109082168886474373197089 x^{52}-914479445566527094599669324 x^{53}+828502745555998906218503832 x^{54}-\frac {3606758093003645324155339152 x^{55}}{5}+603511770853123192467791538 x^{56}-485119509585285628395162576 x^{57}+374593512943317843600698196 x^{58}-277798460089394796889723848 x^{59}+\frac {989058310490690095822896114 x^{60}}{5}-135208860450519457389515112 x^{61}+88685381585824590330757224 x^{62}-55800482090690569716307824 x^{63}+\frac {134663663432573814416170293 x^{64}}{4}-\frac {97339302505596701018770224 x^{65}}{5}+\frac {21569504532490178066659311 x^{66}}{2}-5720500053966970302409071 x^{67}+\frac {11614348594417788189739629 x^{68}}{4}-1409398564020847755666003 x^{69}+\frac {3268857173695411601376612 x^{70}}{5}-289586448945460019391192 x^{71}+122384749256712270099849 x^{72}-49303368020068535591064 x^{73}+18914430223915181446722 x^{74}-\frac {172561788070239874568724 x^{75}}{25}+2393282266977011062653 x^{76}-787400226364730912388 x^{77}+245464847895078057708 x^{78}-72392559119475593544 x^{79}+\frac {201631839342385547403 x^{80}}{10}-5293662917568490696 x^{81}+1307276513180023617 x^{82}-302950588656028082 x^{83}+\frac {131419332012806607 x^{84}}{2}-\frac {66501348729372018 x^{85}}{5}+2503926751715004 x^{86}-436790467844808 x^{87}[phone] x^{88}-10386185673864 x^{89}+\frac {7002807007378 x^{90}}{5}-171200862756 x^{91}[phone] x^{92}-1840869492 x^{93}[phone] x^{94}-\frac {58975224 x^{95}}{5}+\frac {2980131 x^{96}}{4}-38808 x^{97}+\frac {3201 x^{98}}{2}-49 x^{99}+\frac {99 x^{100}}{100}-\frac {x^{101}}{101} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^99*x,x]

[Out]

x^2/2 - 33*x^3 + (4851*x^4)/4 - (156849*x^5)/5 + 627396*x^6 - 10217592*x^7 + 140066157*x^8 - 1654114616*x^9 +
(85600431378*x^10)/5 - 157366449604*x^11 + 1298273209233*x^12 - 9696194317908*x^13 + 66026466069564*x^14 - (20
62057324941768*x^15)/5 + (4750096337812287*x^16)/2 - 12666923567499432*x^17 + 62806829355518017*x^18 - 2905058
91817783026*x^19 + (12572449429225165403*x^20)/10 - 5104603527655330314*x^21 + 19490304378320352108*x^22 - 701
32813684308016488*x^23 + 238292173768273828749*x^24 - (19146258135816088501224*x^25)/25 + 23319160550032415482
26*x^26 - 6736646381120475583764*x^27 + 18488763007525700846649*x^28 - 48264408157576669898532*x^29 + (5998576
44244167183024612*x^30)/5 - 284248449886557530554488*x^31 + (2570079734390957672096829*x^32)/4 - 1386787891870
780679371896*x^33 + (5720500053966970302409071*x^34)/2 - (28206275157871771317939099*x^35)/5 + (42585944846198
556695711973*x^36)/4 - 19237666204653402059452899*x^37 + 33300287699283081927474024*x^38 - 5524663115182515463
2274992*x^39 + (439428796464188236515924114*x^40)/5 - 134109601422466453670901168*x^41 + 196374773511468735732
390996*x^42 - 276016272695076305811040776*x^43 + 372502480574415750374856978*x^44 - (2414047083412492769871166
152*x^45)/5 + 601126348833940887359223192*x^46 - 719077854633508484642475024*x^47 + 82654872964666872011893188
9*x^48 - 913043842041304917057126672*x^49 + (24233705307512968006891237086*x^50)/25 - 989130828878080326811887
228*x^51 + 970109082168886474373197089*x^52 - 914479445566527094599669324*x^53 + 828502745555998906218503832*x
^54 - (3606758093003645324155339152*x^55)/5 + 603511770853123192467791538*x^56 - 485119509585285628395162576*x
^57 + 374593512943317843600698196*x^58 - 277798460089394796889723848*x^59 + (989058310490690095822896114*x^60)
/5 - 135208860450519457389515112*x^61 + 88685381585824590330757224*x^62 - 55800482090690569716307824*x^63 + (1
34663663432573814416170293*x^64)/4 - (97339302505596701018770224*x^65)/5 + (21569504532490178066659311*x^66)/2
- 5720500053966970302409071*x^67 + (11614348594417788189739629*x^68)/4 - 1409398564020847755666003*x^69 + (32
68857173695411601376612*x^70)/5 - 289586448945460019391192*x^71 + 122384749256712270099849*x^72 - 493033680200
68535591064*x^73 + 18914430223915181446722*x^74 - (172561788070239874568724*x^75)/25 + 2393282266977011062653*
x^76 - 787400226364730912388*x^77 + 245464847895078057708*x^78 - 72392559119475593544*x^79 + (2016318393423855
47403*x^80)/10 - 5293662917568490696*x^81 + 1307276513180023617*x^82 - 302950588656028082*x^83 + (131419332012
806607*x^84)/2 - (66501348729372018*x^85)/5 + 2503926751715004*x^86 - 436790467844808*x^87 + 70297408804833*x^
88 - 10386185673864*x^89 + (7002807007378*x^90)/5 - 171200862756*x^91 + 18815553757*x^92 - 1840869492*x^93 + 1
58372676*x^94 - (58975224*x^95)/5 + (2980131*x^96)/4 - 38808*x^97 + (3201*x^98)/2 - 49*x^99 + (99*x^100)/100 -
x^101/101

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(19)=38\).
time = 0.09, size = 502, normalized size = 21.83

method	result	size
gosper	\(\text {Expression too large to display}\)	\(501\)
default	\(\text {Expression too large to display}\)	\(502\)
risch	\(\text {Expression too large to display}\)	\(502\)
parallelrisch	\(\text {Expression too large to display}\)	\(502\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(1-x)^99,x,method=_RETURNVERBOSE)

[Out]

-157366449604*x^11-9696194317908*x^13-2062057324941768/5*x^15-12666923567499432*x^17-1654114616*x^9-10217592*x
^7+4851/4*x^4-156849/5*x^5+627396*x^6-33*x^3+1/2*x^2-289586448945460019391192*x^71+122384749256712270099849*x^
72-49303368020068535591064*x^73+18914430223915181446722*x^74-172561788070239874568724/25*x^75+2393282266977011
062653*x^76-787400226364730912388*x^77+245464847895078057708*x^78-72392559119475593544*x^79+201631839342385547
403/10*x^80-5293662917568490696*x^81+1307276513180023617*x^82-302950588656028082*x^83+131419332012806607/2*x^8
4-66501348729372018/5*x^85+2503926751715004*x^86-436790467844808*x^87+70297408804833*x^88-10386185673864*x^89+
7002807007378/5*x^90-171200862756*x^91+18815553757*x^92-1840869492*x^93+158372676*x^94-58975224/5*x^95+2980131
/4*x^96-38808*x^97+3201/2*x^98-49*x^99+99/100*x^100-1/101*x^101+140066157*x^8+85600431378/5*x^10+1298273209233
*x^12+66026466069564*x^14+4750096337812287/2*x^16+62806829355518017*x^18-290505891817783026*x^19+1257244942922
5165403/10*x^20-5104603527655330314*x^21+19490304378320352108*x^22-70132813684308016488*x^23+23829217376827382
8749*x^24-19146258135816088501224/25*x^25+2331916055003241548226*x^26-6736646381120475583764*x^27+184887630075
25700846649*x^28-48264408157576669898532*x^29+599857644244167183024612/5*x^30-284248449886557530554488*x^31+25
70079734390957672096829/4*x^32-1386787891870780679371896*x^33+5720500053966970302409071/2*x^34-282062751578717
71317939099/5*x^35+42585944846198556695711973/4*x^36-19237666204653402059452899*x^37+3330028769928308192747402
4*x^38-55246631151825154632274992*x^39+439428796464188236515924114/5*x^40-134109601422466453670901168*x^41+196
374773511468735732390996*x^42-276016272695076305811040776*x^43+372502480574415750374856978*x^44-24140470834124
92769871166152/5*x^45+601126348833940887359223192*x^46-719077854633508484642475024*x^47+8265487296466687201189
31889*x^48-913043842041304917057126672*x^49+24233705307512968006891237086/25*x^50-989130828878080326811887228*
x^51+970109082168886474373197089*x^52-914479445566527094599669324*x^53+828502745555998906218503832*x^54-360675
8093003645324155339152/5*x^55+603511770853123192467791538*x^56-485119509585285628395162576*x^57+37459351294331
7843600698196*x^58-277798460089394796889723848*x^59+989058310490690095822896114/5*x^60-13520886045051945738951
5112*x^61+88685381585824590330757224*x^62-55800482090690569716307824*x^63+134663663432573814416170293/4*x^64-9
7339302505596701018770224/5*x^65+21569504532490178066659311/2*x^66-5720500053966970302409071*x^67+116143485944
17788189739629/4*x^68-1409398564020847755666003*x^69+3268857173695411601376612/5*x^70

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).
time = 0.38, size = 501, normalized size = 21.78 \begin {gather*} -\frac {1}{101} \, x^{101} + \frac {99}{100} \, x^{100} - 49 \, x^{99} + \frac {3201}{2} \, x^{98} - 38808 \, x^{97} + \frac {2980131}{4} \, x^{96} - \frac {58975224}{5} \, x^{95} + 158372676 \, x^{94} - 1840869492 \, x^{93} + 18815553757 \, x^{92} - 171200862756 \, x^{91} + \frac {7002807007378}{5} \, x^{90} - 10386185673864 \, x^{89} + 70297408804833 \, x^{88} - 436790467844808 \, x^{87} + 2503926751715004 \, x^{86} - \frac {66501348729372018}{5} \, x^{85} + \frac {131419332012806607}{2} \, x^{84} - 302950588656028082 \, x^{83} + 1307276513180023617 \, x^{82} - 5293662917568490696 \, x^{81} + \frac {201631839342385547403}{10} \, x^{80} - 72392559119475593544 \, x^{79} + 245464847895078057708 \, x^{78} - 787400226364730912388 \, x^{77} + 2393282266977011062653 \, x^{76} - \frac {172561788070239874568724}{25} \, x^{75} + 18914430223915181446722 \, x^{74} - 49303368020068535591064 \, x^{73} + 122384749256712270099849 \, x^{72} - 289586448945460019391192 \, x^{71} + \frac {3268857173695411601376612}{5} \, x^{70} - 1409398564020847755666003 \, x^{69} + \frac {11614348594417788189739629}{4} \, x^{68} - 5720500053966970302409071 \, x^{67} + \frac {21569504532490178066659311}{2} \, x^{66} - \frac {97339302505596701018770224}{5} \, x^{65} + \frac {134663663432573814416170293}{4} \, x^{64} - 55800482090690569716307824 \, x^{63} + 88685381585824590330757224 \, x^{62} - 135208860450519457389515112 \, x^{61} + \frac {989058310490690095822896114}{5} \, x^{60} - 277798460089394796889723848 \, x^{59} + 374593512943317843600698196 \, x^{58} - 485119509585285628395162576 \, x^{57} + 603511770853123192467791538 \, x^{56} - \frac {3606758093003645324155339152}{5} \, x^{55} + 828502745555998906218503832 \, x^{54} - 914479445566527094599669324 \, x^{53} + 970109082168886474373197089 \, x^{52} - 989130828878080326811887228 \, x^{51} + \frac {24233705307512968006891237086}{25} \, x^{50} - 913043842041304917057126672 \, x^{49} + 826548729646668720118931889 \, x^{48} - 719077854633508484642475024 \, x^{47} + 601126348833940887359223192 \, x^{46} - \frac {2414047083412492769871166152}{5} \, x^{45} + 372502480574415750374856978 \, x^{44} - 276016272695076305811040776 \, x^{43} + 196374773511468735732390996 \, x^{42} - 134109601422466453670901168 \, x^{41} + \frac {439428796464188236515924114}{5} \, x^{40} - 55246631151825154632274992 \, x^{39} + 33300287699283081927474024 \, x^{38} - 19237666204653402059452899 \, x^{37} + \frac {42585944846198556695711973}{4} \, x^{36} - \frac {28206275157871771317939099}{5} \, x^{35} + \frac {5720500053966970302409071}{2} \, x^{34} - 1386787891870780679371896 \, x^{33} + \frac {2570079734390957672096829}{4} \, x^{32} - 284248449886557530554488 \, x^{31} + \frac {599857644244167183024612}{5} \, x^{30} - 48264408157576669898532 \, x^{29} + 18488763007525700846649 \, x^{28} - 6736646381120475583764 \, x^{27} + 2331916055003241548226 \, x^{26} - \frac {19146258135816088501224}{25} \, x^{25} + 238292173768273828749 \, x^{24} - 70132813684308016488 \, x^{23} + 19490304378320352108 \, x^{22} - 5104603527655330314 \, x^{21} + \frac {12572449429225165403}{10} \, x^{20} - 290505891817783026 \, x^{19} + 62806829355518017 \, x^{18} - 12666923567499432 \, x^{17} + \frac {4750096337812287}{2} \, x^{16} - \frac {2062057324941768}{5} \, x^{15} + 66026466069564 \, x^{14} - 9696194317908 \, x^{13} + 1298273209233 \, x^{12} - 157366449604 \, x^{11} + \frac {85600431378}{5} \, x^{10} - 1654114616 \, x^{9} + 140066157 \, x^{8} - 10217592 \, x^{7} + 627396 \, x^{6} - \frac {156849}{5} \, x^{5} + \frac {4851}{4} \, x^{4} - 33 \, x^{3} + \frac {1}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1-x)^99,x, algorithm="maxima")

[Out]

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131/4*x^96 - 58975224/5*x^95 + 15837267
6*x^94 - 1840869492*x^93 + 18815553757*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 +
70297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 66501348729372018/5*x^85 + 1314193320128
06607/2*x^84 - 302950588656028082*x^83 + 1307276513180023617*x^82 - 5293662917568490696*x^81 + 201631839342385
547403/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 787400226364730912388*x^77 + 2393282
266977011062653*x^76 - 172561788070239874568724/25*x^75 + 18914430223915181446722*x^74 - 493033680200685355910
64*x^73 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 3268857173695411601376612/5*x^70 - 1
409398564020847755666003*x^69 + 11614348594417788189739629/4*x^68 - 5720500053966970302409071*x^67 + 215695045
32490178066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 134663663432573814416170293/4*x^64 - 5580048209
0690569716307824*x^63 + 88685381585824590330757224*x^62 - 135208860450519457389515112*x^61 + 98905831049069009
5822896114/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x^58 - 48511950958528562839
5162576*x^57 + 603511770853123192467791538*x^56 - 3606758093003645324155339152/5*x^55 + 8285027455559989062185
03832*x^54 - 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989130828878080326811887228
*x^51 + 24233705307512968006891237086/25*x^50 - 913043842041304917057126672*x^49 + 826548729646668720118931889
*x^48 - 719077854633508484642475024*x^47 + 601126348833940887359223192*x^46 - 2414047083412492769871166152/5*x
^45 + 372502480574415750374856978*x^44 - 276016272695076305811040776*x^43 + 196374773511468735732390996*x^42 -
134109601422466453670901168*x^41 + 439428796464188236515924114/5*x^40 - 55246631151825154632274992*x^39 + 333
00287699283081927474024*x^38 - 19237666204653402059452899*x^37 + 42585944846198556695711973/4*x^36 - 282062751
57871771317939099/5*x^35 + 5720500053966970302409071/2*x^34 - 1386787891870780679371896*x^33 + 257007973439095
7672096829/4*x^32 - 284248449886557530554488*x^31 + 599857644244167183024612/5*x^30 - 48264408157576669898532*
x^29 + 18488763007525700846649*x^28 - 6736646381120475583764*x^27 + 2331916055003241548226*x^26 - 191462581358
16088501224/25*x^25 + 238292173768273828749*x^24 - 70132813684308016488*x^23 + 19490304378320352108*x^22 - 510
4603527655330314*x^21 + 12572449429225165403/10*x^20 - 290505891817783026*x^19 + 62806829355518017*x^18 - 1266
6923567499432*x^17 + 4750096337812287/2*x^16 - 2062057324941768/5*x^15 + 66026466069564*x^14 - 9696194317908*x
^13 + 1298273209233*x^12 - 157366449604*x^11 + 85600431378/5*x^10 - 1654114616*x^9 + 140066157*x^8 - 10217592*
x^7 + 627396*x^6 - 156849/5*x^5 + 4851/4*x^4 - 33*x^3 + 1/2*x^2

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).
time = 0.56, size = 501, normalized size = 21.78 \begin {gather*} -\frac {1}{101} \, x^{101} + \frac {99}{100} \, x^{100} - 49 \, x^{99} + \frac {3201}{2} \, x^{98} - 38808 \, x^{97} + \frac {2980131}{4} \, x^{96} - \frac {58975224}{5} \, x^{95} + 158372676 \, x^{94} - 1840869492 \, x^{93} + 18815553757 \, x^{92} - 171200862756 \, x^{91} + \frac {7002807007378}{5} \, x^{90} - 10386185673864 \, x^{89} + 70297408804833 \, x^{88} - 436790467844808 \, x^{87} + 2503926751715004 \, x^{86} - \frac {66501348729372018}{5} \, x^{85} + \frac {131419332012806607}{2} \, x^{84} - 302950588656028082 \, x^{83} + 1307276513180023617 \, x^{82} - 5293662917568490696 \, x^{81} + \frac {201631839342385547403}{10} \, x^{80} - 72392559119475593544 \, x^{79} + 245464847895078057708 \, x^{78} - 787400226364730912388 \, x^{77} + 2393282266977011062653 \, x^{76} - \frac {172561788070239874568724}{25} \, x^{75} + 18914430223915181446722 \, x^{74} - 49303368020068535591064 \, x^{73} + 122384749256712270099849 \, x^{72} - 289586448945460019391192 \, x^{71} + \frac {3268857173695411601376612}{5} \, x^{70} - 1409398564020847755666003 \, x^{69} + \frac {11614348594417788189739629}{4} \, x^{68} - 5720500053966970302409071 \, x^{67} + \frac {21569504532490178066659311}{2} \, x^{66} - \frac {97339302505596701018770224}{5} \, x^{65} + \frac {134663663432573814416170293}{4} \, x^{64} - 55800482090690569716307824 \, x^{63} + 88685381585824590330757224 \, x^{62} - 135208860450519457389515112 \, x^{61} + \frac {989058310490690095822896114}{5} \, x^{60} - 277798460089394796889723848 \, x^{59} + 374593512943317843600698196 \, x^{58} - 485119509585285628395162576 \, x^{57} + 603511770853123192467791538 \, x^{56} - \frac {3606758093003645324155339152}{5} \, x^{55} + 828502745555998906218503832 \, x^{54} - 914479445566527094599669324 \, x^{53} + 970109082168886474373197089 \, x^{52} - 989130828878080326811887228 \, x^{51} + \frac {24233705307512968006891237086}{25} \, x^{50} - 913043842041304917057126672 \, x^{49} + 826548729646668720118931889 \, x^{48} - 719077854633508484642475024 \, x^{47} + 601126348833940887359223192 \, x^{46} - \frac {2414047083412492769871166152}{5} \, x^{45} + 372502480574415750374856978 \, x^{44} - 276016272695076305811040776 \, x^{43} + 196374773511468735732390996 \, x^{42} - 134109601422466453670901168 \, x^{41} + \frac {439428796464188236515924114}{5} \, x^{40} - 55246631151825154632274992 \, x^{39} + 33300287699283081927474024 \, x^{38} - 19237666204653402059452899 \, x^{37} + \frac {42585944846198556695711973}{4} \, x^{36} - \frac {28206275157871771317939099}{5} \, x^{35} + \frac {5720500053966970302409071}{2} \, x^{34} - 1386787891870780679371896 \, x^{33} + \frac {2570079734390957672096829}{4} \, x^{32} - 284248449886557530554488 \, x^{31} + \frac {599857644244167183024612}{5} \, x^{30} - 48264408157576669898532 \, x^{29} + 18488763007525700846649 \, x^{28} - 6736646381120475583764 \, x^{27} + 2331916055003241548226 \, x^{26} - \frac {19146258135816088501224}{25} \, x^{25} + 238292173768273828749 \, x^{24} - 70132813684308016488 \, x^{23} + 19490304378320352108 \, x^{22} - 5104603527655330314 \, x^{21} + \frac {12572449429225165403}{10} \, x^{20} - 290505891817783026 \, x^{19} + 62806829355518017 \, x^{18} - 12666923567499432 \, x^{17} + \frac {4750096337812287}{2} \, x^{16} - \frac {2062057324941768}{5} \, x^{15} + 66026466069564 \, x^{14} - 9696194317908 \, x^{13} + 1298273209233 \, x^{12} - 157366449604 \, x^{11} + \frac {85600431378}{5} \, x^{10} - 1654114616 \, x^{9} + 140066157 \, x^{8} - 10217592 \, x^{7} + 627396 \, x^{6} - \frac {156849}{5} \, x^{5} + \frac {4851}{4} \, x^{4} - 33 \, x^{3} + \frac {1}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1-x)^99,x, algorithm="fricas")

[Out]

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131/4*x^96 - 58975224/5*x^95 + 15837267
6*x^94 - 1840869492*x^93 + 18815553757*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 +
70297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 66501348729372018/5*x^85 + 1314193320128
06607/2*x^84 - 302950588656028082*x^83 + 1307276513180023617*x^82 - 5293662917568490696*x^81 + 201631839342385
547403/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 787400226364730912388*x^77 + 2393282
266977011062653*x^76 - 172561788070239874568724/25*x^75 + 18914430223915181446722*x^74 - 493033680200685355910
64*x^73 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 3268857173695411601376612/5*x^70 - 1
409398564020847755666003*x^69 + 11614348594417788189739629/4*x^68 - 5720500053966970302409071*x^67 + 215695045
32490178066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 134663663432573814416170293/4*x^64 - 5580048209
0690569716307824*x^63 + 88685381585824590330757224*x^62 - 135208860450519457389515112*x^61 + 98905831049069009
5822896114/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x^58 - 48511950958528562839
5162576*x^57 + 603511770853123192467791538*x^56 - 3606758093003645324155339152/5*x^55 + 8285027455559989062185
03832*x^54 - 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989130828878080326811887228
*x^51 + 24233705307512968006891237086/25*x^50 - 913043842041304917057126672*x^49 + 826548729646668720118931889
*x^48 - 719077854633508484642475024*x^47 + 601126348833940887359223192*x^46 - 2414047083412492769871166152/5*x
^45 + 372502480574415750374856978*x^44 - 276016272695076305811040776*x^43 + 196374773511468735732390996*x^42 -
134109601422466453670901168*x^41 + 439428796464188236515924114/5*x^40 - 55246631151825154632274992*x^39 + 333
00287699283081927474024*x^38 - 19237666204653402059452899*x^37 + 42585944846198556695711973/4*x^36 - 282062751
57871771317939099/5*x^35 + 5720500053966970302409071/2*x^34 - 1386787891870780679371896*x^33 + 257007973439095
7672096829/4*x^32 - 284248449886557530554488*x^31 + 599857644244167183024612/5*x^30 - 48264408157576669898532*
x^29 + 18488763007525700846649*x^28 - 6736646381120475583764*x^27 + 2331916055003241548226*x^26 - 191462581358
16088501224/25*x^25 + 238292173768273828749*x^24 - 70132813684308016488*x^23 + 19490304378320352108*x^22 - 510
4603527655330314*x^21 + 12572449429225165403/10*x^20 - 290505891817783026*x^19 + 62806829355518017*x^18 - 1266
6923567499432*x^17 + 4750096337812287/2*x^16 - 2062057324941768/5*x^15 + 66026466069564*x^14 - 9696194317908*x
^13 + 1298273209233*x^12 - 157366449604*x^11 + 85600431378/5*x^10 - 1654114616*x^9 + 140066157*x^8 - 10217592*
x^7 + 627396*x^6 - 156849/5*x^5 + 4851/4*x^4 - 33*x^3 + 1/2*x^2

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 561 vs. \(2 (12) = 24\).
time = 0.10, size = 561, normalized size = 24.39 \begin {gather*} - \frac {x^{101}}{101} + \frac {99 x^{100}}{100} - 49 x^{99} + \frac {3201 x^{98}}{2} - 38808 x^{97} + \frac {2980131 x^{96}}{4} - \frac {58975224 x^{95}}{5} + 158372676 x^{94} - 1840869492 x^{93} + 18815553757 x^{92} - 171200862756 x^{91} + \frac {7002807007378 x^{90}}{5} - 10386185673864 x^{89} + 70297408804833 x^{88} - 436790467844808 x^{87} + 2503926751715004 x^{86} - \frac {66501348729372018 x^{85}}{5} + \frac {131419332012806607 x^{84}}{2} - 302950588656028082 x^{83} + 1307276513180023617 x^{82} - 5293662917568490696 x^{81} + \frac {201631839342385547403 x^{80}}{10} - 72392559119475593544 x^{79} + 245464847895078057708 x^{78} - 787400226364730912388 x^{77} + 2393282266977011062653 x^{76} - \frac {172561788070239874568724 x^{75}}{25} + 18914430223915181446722 x^{74} - 49303368020068535591064 x^{73} + 122384749256712270099849 x^{72} - 289586448945460019391192 x^{71} + \frac {3268857173695411601376612 x^{70}}{5} - 1409398564020847755666003 x^{69} + \frac {11614348594417788189739629 x^{68}}{4} - 5720500053966970302409071 x^{67} + \frac {21569504532490178066659311 x^{66}}{2} - \frac {97339302505596701018770224 x^{65}}{5} + \frac {134663663432573814416170293 x^{64}}{4} - 55800482090690569716307824 x^{63} + 88685381585824590330757224 x^{62} - 135208860450519457389515112 x^{61} + \frac {989058310490690095822896114 x^{60}}{5} - 277798460089394796889723848 x^{59} + 374593512943317843600698196 x^{58} - 485119509585285628395162576 x^{57} + 603511770853123192467791538 x^{56} - \frac {3606758093003645324155339152 x^{55}}{5} + 828502745555998906218503832 x^{54} - 914479445566527094599669324 x^{53} + 970109082168886474373197089 x^{52} - 989130828878080326811887228 x^{51} + \frac {24233705307512968006891237086 x^{50}}{25} - 913043842041304917057126672 x^{49} + 826548729646668720118931889 x^{48} - 719077854633508484642475024 x^{47} + 601126348833940887359223192 x^{46} - \frac {2414047083412492769871166152 x^{45}}{5} + 372502480574415750374856978 x^{44} - 276016272695076305811040776 x^{43} + 196374773511468735732390996 x^{42} - 134109601422466453670901168 x^{41} + \frac {439428796464188236515924114 x^{40}}{5} - 55246631151825154632274992 x^{39} + 33300287699283081927474024 x^{38} - 19237666204653402059452899 x^{37} + \frac {42585944846198556695711973 x^{36}}{4} - \frac {28206275157871771317939099 x^{35}}{5} + \frac {5720500053966970302409071 x^{34}}{2} - 1386787891870780679371896 x^{33} + \frac {2570079734390957672096829 x^{32}}{4} - 284248449886557530554488 x^{31} + \frac {599857644244167183024612 x^{30}}{5} - 48264408157576669898532 x^{29} + 18488763007525700846649 x^{28} - 6736646381120475583764 x^{27} + 2331916055003241548226 x^{26} - \frac {19146258135816088501224 x^{25}}{25} + 238292173768273828749 x^{24} - 70132813684308016488 x^{23} + 19490304378320352108 x^{22} - 5104603527655330314 x^{21} + \frac {12572449429225165403 x^{20}}{10} - 290505891817783026 x^{19} + 62806829355518017 x^{18} - 12666923567499432 x^{17} + \frac {4750096337812287 x^{16}}{2} - \frac {2062057324941768 x^{15}}{5} + 66026466069564 x^{14} - 9696194317908 x^{13} + 1298273209233 x^{12} - 157366449604 x^{11} + \frac {85600431378 x^{10}}{5} - 1654114616 x^{9} + 140066157 x^{8} - 10217592 x^{7} + 627396 x^{6} - \frac {156849 x^{5}}{5} + \frac {4851 x^{4}}{4} - 33 x^{3} + \frac {x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1-x)**99,x)

[Out]

-x**101/101 + 99*x**100/100 - 49*x**99 + 3201*x**98/2 - 38808*x**97 + 2980131*x**96/4 - 58975224*x**95/5 + 158
372676*x**94 - 1840869492*x**93 + 18815553757*x**92 - 171200862756*x**91 + 7002807007378*x**90/5 - 10386185673
864*x**89 + 70297408804833*x**88 - 436790467844808*x**87 + 2503926751715004*x**86 - 66501348729372018*x**85/5
+ 131419332012806607*x**84/2 - 302950588656028082*x**83 + 1307276513180023617*x**82 - 5293662917568490696*x**8
1 + 201631839342385547403*x**80/10 - 72392559119475593544*x**79 + 245464847895078057708*x**78 - 78740022636473
0912388*x**77 + 2393282266977011062653*x**76 - 172561788070239874568724*x**75/25 + 18914430223915181446722*x**
74 - 49303368020068535591064*x**73 + 122384749256712270099849*x**72 - 289586448945460019391192*x**71 + 3268857
173695411601376612*x**70/5 - 1409398564020847755666003*x**69 + 11614348594417788189739629*x**68/4 - 5720500053
966970302409071*x**67 + 21569504532490178066659311*x**66/2 - 97339302505596701018770224*x**65/5 + 134663663432
573814416170293*x**64/4 - 55800482090690569716307824*x**63 + 88685381585824590330757224*x**62 - 13520886045051
9457389515112*x**61 + 989058310490690095822896114*x**60/5 - 277798460089394796889723848*x**59 + 37459351294331
7843600698196*x**58 - 485119509585285628395162576*x**57 + 603511770853123192467791538*x**56 - 3606758093003645
324155339152*x**55/5 + 828502745555998906218503832*x**54 - 914479445566527094599669324*x**53 + 970109082168886
474373197089*x**52 - 989130828878080326811887228*x**51 + 24233705307512968006891237086*x**50/25 - 913043842041
304917057126672*x**49 + 826548729646668720118931889*x**48 - 719077854633508484642475024*x**47 + 60112634883394
0887359223192*x**46 - 2414047083412492769871166152*x**45/5 + 372502480574415750374856978*x**44 - 2760162726950
76305811040776*x**43 + 196374773511468735732390996*x**42 - 134109601422466453670901168*x**41 + 439428796464188
236515924114*x**40/5 - 55246631151825154632274992*x**39 + 33300287699283081927474024*x**38 - 19237666204653402
059452899*x**37 + 42585944846198556695711973*x**36/4 - 28206275157871771317939099*x**35/5 + 572050005396697030
2409071*x**34/2 - 1386787891870780679371896*x**33 + 2570079734390957672096829*x**32/4 - 2842484498865575305544
88*x**31 + 599857644244167183024612*x**30/5 - 48264408157576669898532*x**29 + 18488763007525700846649*x**28 -
6736646381120475583764*x**27 + 2331916055003241548226*x**26 - 19146258135816088501224*x**25/25 + 2382921737682
73828749*x**24 - 70132813684308016488*x**23 + 19490304378320352108*x**22 - 5104603527655330314*x**21 + 1257244
9429225165403*x**20/10 - 290505891817783026*x**19 + 62806829355518017*x**18 - 12666923567499432*x**17 + 475009
6337812287*x**16/2 - 2062057324941768*x**15/5 + 66026466069564*x**14 - 9696194317908*x**13 + 1298273209233*x**
12 - 157366449604*x**11 + 85600431378*x**10/5 - 1654114616*x**9 + 140066157*x**8 - 10217592*x**7 + 627396*x**6
- 156849*x**5/5 + 4851*x**4/4 - 33*x**3 + x**2/2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 501 vs. \(2 (15) = 30\).
time = 0.44, size = 501, normalized size = 21.78 \begin {gather*} -\frac {1}{101} \, x^{101} + \frac {99}{100} \, x^{100} - 49 \, x^{99} + \frac {3201}{2} \, x^{98} - 38808 \, x^{97} + \frac {2980131}{4} \, x^{96} - \frac {58975224}{5} \, x^{95} + 158372676 \, x^{94} - 1840869492 \, x^{93} + 18815553757 \, x^{92} - 171200862756 \, x^{91} + \frac {7002807007378}{5} \, x^{90} - 10386185673864 \, x^{89} + 70297408804833 \, x^{88} - 436790467844808 \, x^{87} + 2503926751715004 \, x^{86} - \frac {66501348729372018}{5} \, x^{85} + \frac {131419332012806607}{2} \, x^{84} - 302950588656028082 \, x^{83} + 1307276513180023617 \, x^{82} - 5293662917568490696 \, x^{81} + \frac {201631839342385547403}{10} \, x^{80} - 72392559119475593544 \, x^{79} + 245464847895078057708 \, x^{78} - 787400226364730912388 \, x^{77} + 2393282266977011062653 \, x^{76} - \frac {172561788070239874568724}{25} \, x^{75} + 18914430223915181446722 \, x^{74} - 49303368020068535591064 \, x^{73} + 122384749256712270099849 \, x^{72} - 289586448945460019391192 \, x^{71} + \frac {3268857173695411601376612}{5} \, x^{70} - 1409398564020847755666003 \, x^{69} + \frac {11614348594417788189739629}{4} \, x^{68} - 5720500053966970302409071 \, x^{67} + \frac {21569504532490178066659311}{2} \, x^{66} - \frac {97339302505596701018770224}{5} \, x^{65} + \frac {134663663432573814416170293}{4} \, x^{64} - 55800482090690569716307824 \, x^{63} + 88685381585824590330757224 \, x^{62} - 135208860450519457389515112 \, x^{61} + \frac {989058310490690095822896114}{5} \, x^{60} - 277798460089394796889723848 \, x^{59} + 374593512943317843600698196 \, x^{58} - 485119509585285628395162576 \, x^{57} + 603511770853123192467791538 \, x^{56} - \frac {3606758093003645324155339152}{5} \, x^{55} + 828502745555998906218503832 \, x^{54} - 914479445566527094599669324 \, x^{53} + 970109082168886474373197089 \, x^{52} - 989130828878080326811887228 \, x^{51} + \frac {24233705307512968006891237086}{25} \, x^{50} - 913043842041304917057126672 \, x^{49} + 826548729646668720118931889 \, x^{48} - 719077854633508484642475024 \, x^{47} + 601126348833940887359223192 \, x^{46} - \frac {2414047083412492769871166152}{5} \, x^{45} + 372502480574415750374856978 \, x^{44} - 276016272695076305811040776 \, x^{43} + 196374773511468735732390996 \, x^{42} - 134109601422466453670901168 \, x^{41} + \frac {439428796464188236515924114}{5} \, x^{40} - 55246631151825154632274992 \, x^{39} + 33300287699283081927474024 \, x^{38} - 19237666204653402059452899 \, x^{37} + \frac {42585944846198556695711973}{4} \, x^{36} - \frac {28206275157871771317939099}{5} \, x^{35} + \frac {5720500053966970302409071}{2} \, x^{34} - 1386787891870780679371896 \, x^{33} + \frac {2570079734390957672096829}{4} \, x^{32} - 284248449886557530554488 \, x^{31} + \frac {599857644244167183024612}{5} \, x^{30} - 48264408157576669898532 \, x^{29} + 18488763007525700846649 \, x^{28} - 6736646381120475583764 \, x^{27} + 2331916055003241548226 \, x^{26} - \frac {19146258135816088501224}{25} \, x^{25} + 238292173768273828749 \, x^{24} - 70132813684308016488 \, x^{23} + 19490304378320352108 \, x^{22} - 5104603527655330314 \, x^{21} + \frac {12572449429225165403}{10} \, x^{20} - 290505891817783026 \, x^{19} + 62806829355518017 \, x^{18} - 12666923567499432 \, x^{17} + \frac {4750096337812287}{2} \, x^{16} - \frac {2062057324941768}{5} \, x^{15} + 66026466069564 \, x^{14} - 9696194317908 \, x^{13} + 1298273209233 \, x^{12} - 157366449604 \, x^{11} + \frac {85600431378}{5} \, x^{10} - 1654114616 \, x^{9} + 140066157 \, x^{8} - 10217592 \, x^{7} + 627396 \, x^{6} - \frac {156849}{5} \, x^{5} + \frac {4851}{4} \, x^{4} - 33 \, x^{3} + \frac {1}{2} \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(1-x)^99,x, algorithm="giac")

[Out]

-1/101*x^101 + 99/100*x^100 - 49*x^99 + 3201/2*x^98 - 38808*x^97 + 2980131/4*x^96 - 58975224/5*x^95 + 15837267
6*x^94 - 1840869492*x^93 + 18815553757*x^92 - 171200862756*x^91 + 7002807007378/5*x^90 - 10386185673864*x^89 +
70297408804833*x^88 - 436790467844808*x^87 + 2503926751715004*x^86 - 66501348729372018/5*x^85 + 1314193320128
06607/2*x^84 - 302950588656028082*x^83 + 1307276513180023617*x^82 - 5293662917568490696*x^81 + 201631839342385
547403/10*x^80 - 72392559119475593544*x^79 + 245464847895078057708*x^78 - 787400226364730912388*x^77 + 2393282
266977011062653*x^76 - 172561788070239874568724/25*x^75 + 18914430223915181446722*x^74 - 493033680200685355910
64*x^73 + 122384749256712270099849*x^72 - 289586448945460019391192*x^71 + 3268857173695411601376612/5*x^70 - 1
409398564020847755666003*x^69 + 11614348594417788189739629/4*x^68 - 5720500053966970302409071*x^67 + 215695045
32490178066659311/2*x^66 - 97339302505596701018770224/5*x^65 + 134663663432573814416170293/4*x^64 - 5580048209
0690569716307824*x^63 + 88685381585824590330757224*x^62 - 135208860450519457389515112*x^61 + 98905831049069009
5822896114/5*x^60 - 277798460089394796889723848*x^59 + 374593512943317843600698196*x^58 - 48511950958528562839
5162576*x^57 + 603511770853123192467791538*x^56 - 3606758093003645324155339152/5*x^55 + 8285027455559989062185
03832*x^54 - 914479445566527094599669324*x^53 + 970109082168886474373197089*x^52 - 989130828878080326811887228
*x^51 + 24233705307512968006891237086/25*x^50 - 913043842041304917057126672*x^49 + 826548729646668720118931889
*x^48 - 719077854633508484642475024*x^47 + 601126348833940887359223192*x^46 - 2414047083412492769871166152/5*x
^45 + 372502480574415750374856978*x^44 - 276016272695076305811040776*x^43 + 196374773511468735732390996*x^42 -
134109601422466453670901168*x^41 + 439428796464188236515924114/5*x^40 - 55246631151825154632274992*x^39 + 333
00287699283081927474024*x^38 - 19237666204653402059452899*x^37 + 42585944846198556695711973/4*x^36 - 282062751
57871771317939099/5*x^35 + 5720500053966970302409071/2*x^34 - 1386787891870780679371896*x^33 + 257007973439095
7672096829/4*x^32 - 284248449886557530554488*x^31 + 599857644244167183024612/5*x^30 - 48264408157576669898532*
x^29 + 18488763007525700846649*x^28 - 6736646381120475583764*x^27 + 2331916055003241548226*x^26 - 191462581358
16088501224/25*x^25 + 238292173768273828749*x^24 - 70132813684308016488*x^23 + 19490304378320352108*x^22 - 510
4603527655330314*x^21 + 12572449429225165403/10*x^20 - 290505891817783026*x^19 + 62806829355518017*x^18 - 1266
6923567499432*x^17 + 4750096337812287/2*x^16 - 2062057324941768/5*x^15 + 66026466069564*x^14 - 9696194317908*x
^13 + 1298273209233*x^12 - 157366449604*x^11 + 85600431378/5*x^10 - 1654114616*x^9 + 140066157*x^8 - 10217592*
x^7 + 627396*x^6 - 156849/5*x^5 + 4851/4*x^4 - 33*x^3 + 1/2*x^2

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Mupad [B]
time = 0.12, size = 12, normalized size = 0.52 \begin {gather*} -\frac {\left (100\,x+1\right )\,{\left (x-1\right )}^{100}}{10100} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x*(x - 1)^99,x)

[Out]

-((100*x + 1)*(x - 1)^100)/10100

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Chatgpt [F] Failed to verify
time = 1.00, size = 20, normalized size = 0.87 \begin {gather*} \frac {\left (1-x \right )^{101}}{10100}-\frac {x \left (1-x \right )^{100}}{100} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(x*(1-x)^99,x)

[Out]

1/10100*(1-x)^101-1/100*x*(1-x)^100

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